3.1318 \(\int \frac{(c+d x)^{10}}{(a+b x)^7} \, dx\)

Optimal. Leaf size=262 \[ \frac{10 d^9 (a+b x)^3 (b c-a d)}{3 b^{11}}+\frac{45 d^8 (a+b x)^2 (b c-a d)^2}{2 b^{11}}+\frac{210 d^6 (b c-a d)^4 \log (a+b x)}{b^{11}}-\frac{252 d^5 (b c-a d)^5}{b^{11} (a+b x)}-\frac{105 d^4 (b c-a d)^6}{b^{11} (a+b x)^2}-\frac{40 d^3 (b c-a d)^7}{b^{11} (a+b x)^3}-\frac{45 d^2 (b c-a d)^8}{4 b^{11} (a+b x)^4}-\frac{2 d (b c-a d)^9}{b^{11} (a+b x)^5}-\frac{(b c-a d)^{10}}{6 b^{11} (a+b x)^6}+\frac{d^{10} (a+b x)^4}{4 b^{11}}+\frac{120 d^7 x (b c-a d)^3}{b^{10}} \]

[Out]

(120*d^7*(b*c - a*d)^3*x)/b^10 - (b*c - a*d)^10/(6*b^11*(a + b*x)^6) - (2*d*(b*c
- a*d)^9)/(b^11*(a + b*x)^5) - (45*d^2*(b*c - a*d)^8)/(4*b^11*(a + b*x)^4) - (4
0*d^3*(b*c - a*d)^7)/(b^11*(a + b*x)^3) - (105*d^4*(b*c - a*d)^6)/(b^11*(a + b*x
)^2) - (252*d^5*(b*c - a*d)^5)/(b^11*(a + b*x)) + (45*d^8*(b*c - a*d)^2*(a + b*x
)^2)/(2*b^11) + (10*d^9*(b*c - a*d)*(a + b*x)^3)/(3*b^11) + (d^10*(a + b*x)^4)/(
4*b^11) + (210*d^6*(b*c - a*d)^4*Log[a + b*x])/b^11

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Rubi [A]  time = 0.896807, antiderivative size = 262, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{10 d^9 (a+b x)^3 (b c-a d)}{3 b^{11}}+\frac{45 d^8 (a+b x)^2 (b c-a d)^2}{2 b^{11}}+\frac{210 d^6 (b c-a d)^4 \log (a+b x)}{b^{11}}-\frac{252 d^5 (b c-a d)^5}{b^{11} (a+b x)}-\frac{105 d^4 (b c-a d)^6}{b^{11} (a+b x)^2}-\frac{40 d^3 (b c-a d)^7}{b^{11} (a+b x)^3}-\frac{45 d^2 (b c-a d)^8}{4 b^{11} (a+b x)^4}-\frac{2 d (b c-a d)^9}{b^{11} (a+b x)^5}-\frac{(b c-a d)^{10}}{6 b^{11} (a+b x)^6}+\frac{d^{10} (a+b x)^4}{4 b^{11}}+\frac{120 d^7 x (b c-a d)^3}{b^{10}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^10/(a + b*x)^7,x]

[Out]

(120*d^7*(b*c - a*d)^3*x)/b^10 - (b*c - a*d)^10/(6*b^11*(a + b*x)^6) - (2*d*(b*c
- a*d)^9)/(b^11*(a + b*x)^5) - (45*d^2*(b*c - a*d)^8)/(4*b^11*(a + b*x)^4) - (4
0*d^3*(b*c - a*d)^7)/(b^11*(a + b*x)^3) - (105*d^4*(b*c - a*d)^6)/(b^11*(a + b*x
)^2) - (252*d^5*(b*c - a*d)^5)/(b^11*(a + b*x)) + (45*d^8*(b*c - a*d)^2*(a + b*x
)^2)/(2*b^11) + (10*d^9*(b*c - a*d)*(a + b*x)^3)/(3*b^11) + (d^10*(a + b*x)^4)/(
4*b^11) + (210*d^6*(b*c - a*d)^4*Log[a + b*x])/b^11

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Rubi in Sympy [A]  time = 134.581, size = 243, normalized size = 0.93 \[ - \frac{120 d^{7} x \left (a d - b c\right )^{3}}{b^{10}} + \frac{d^{10} \left (a + b x\right )^{4}}{4 b^{11}} - \frac{10 d^{9} \left (a + b x\right )^{3} \left (a d - b c\right )}{3 b^{11}} + \frac{45 d^{8} \left (a + b x\right )^{2} \left (a d - b c\right )^{2}}{2 b^{11}} + \frac{210 d^{6} \left (a d - b c\right )^{4} \log{\left (a + b x \right )}}{b^{11}} + \frac{252 d^{5} \left (a d - b c\right )^{5}}{b^{11} \left (a + b x\right )} - \frac{105 d^{4} \left (a d - b c\right )^{6}}{b^{11} \left (a + b x\right )^{2}} + \frac{40 d^{3} \left (a d - b c\right )^{7}}{b^{11} \left (a + b x\right )^{3}} - \frac{45 d^{2} \left (a d - b c\right )^{8}}{4 b^{11} \left (a + b x\right )^{4}} + \frac{2 d \left (a d - b c\right )^{9}}{b^{11} \left (a + b x\right )^{5}} - \frac{\left (a d - b c\right )^{10}}{6 b^{11} \left (a + b x\right )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**10/(b*x+a)**7,x)

[Out]

-120*d**7*x*(a*d - b*c)**3/b**10 + d**10*(a + b*x)**4/(4*b**11) - 10*d**9*(a + b
*x)**3*(a*d - b*c)/(3*b**11) + 45*d**8*(a + b*x)**2*(a*d - b*c)**2/(2*b**11) + 2
10*d**6*(a*d - b*c)**4*log(a + b*x)/b**11 + 252*d**5*(a*d - b*c)**5/(b**11*(a +
b*x)) - 105*d**4*(a*d - b*c)**6/(b**11*(a + b*x)**2) + 40*d**3*(a*d - b*c)**7/(b
**11*(a + b*x)**3) - 45*d**2*(a*d - b*c)**8/(4*b**11*(a + b*x)**4) + 2*d*(a*d -
b*c)**9/(b**11*(a + b*x)**5) - (a*d - b*c)**10/(6*b**11*(a + b*x)**6)

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Mathematica [A]  time = 0.387859, size = 265, normalized size = 1.01 \[ \frac{6 b^2 d^8 x^2 \left (28 a^2 d^2-70 a b c d+45 b^2 c^2\right )+12 b d^7 x \left (-84 a^3 d^3+280 a^2 b c d^2-315 a b^2 c^2 d+120 b^3 c^3\right )+4 b^3 d^9 x^3 (10 b c-7 a d)+2520 d^6 (b c-a d)^4 \log (a+b x)+\frac{3024 d^5 (a d-b c)^5}{a+b x}-\frac{1260 d^4 (b c-a d)^6}{(a+b x)^2}+\frac{480 d^3 (a d-b c)^7}{(a+b x)^3}-\frac{135 d^2 (b c-a d)^8}{(a+b x)^4}+\frac{24 d (a d-b c)^9}{(a+b x)^5}-\frac{2 (b c-a d)^{10}}{(a+b x)^6}+3 b^4 d^{10} x^4}{12 b^{11}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^10/(a + b*x)^7,x]

[Out]

(12*b*d^7*(120*b^3*c^3 - 315*a*b^2*c^2*d + 280*a^2*b*c*d^2 - 84*a^3*d^3)*x + 6*b
^2*d^8*(45*b^2*c^2 - 70*a*b*c*d + 28*a^2*d^2)*x^2 + 4*b^3*d^9*(10*b*c - 7*a*d)*x
^3 + 3*b^4*d^10*x^4 - (2*(b*c - a*d)^10)/(a + b*x)^6 + (24*d*(-(b*c) + a*d)^9)/(
a + b*x)^5 - (135*d^2*(b*c - a*d)^8)/(a + b*x)^4 + (480*d^3*(-(b*c) + a*d)^7)/(a
+ b*x)^3 - (1260*d^4*(b*c - a*d)^6)/(a + b*x)^2 + (3024*d^5*(-(b*c) + a*d)^5)/(
a + b*x) + 2520*d^6*(b*c - a*d)^4*Log[a + b*x])/(12*b^11)

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Maple [B]  time = 0.028, size = 1222, normalized size = 4.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^10/(b*x+a)^7,x)

[Out]

2/b^11*d^10/(b*x+a)^5*a^9-2/b^2*d/(b*x+a)^5*c^9+210/b^11*d^10*ln(b*x+a)*a^4+210/
b^7*d^6*ln(b*x+a)*c^4+45/2*d^8/b^7*x^2*c^2-84*d^10/b^10*a^3*x+120*d^7/b^7*c^3*x-
7/3*d^10/b^8*x^3*a+10/3*d^9/b^7*x^3*c+14*d^10/b^9*x^2*a^2-105/b^5*d^4/(b*x+a)^2*
c^6+252/b^11*d^10/(b*x+a)*a^5-252/b^6*d^5/(b*x+a)*c^5-105/b^11*d^10/(b*x+a)^2*a^
6-1/6/b^11/(b*x+a)^6*a^10*d^10-45/4/b^11*d^10/(b*x+a)^4*a^8-45/4/b^3*d^2/(b*x+a)
^4*c^8+40/b^11*d^10/(b*x+a)^3*a^7-40/b^4*d^3/(b*x+a)^3*c^7-1260/b^10*d^9/(b*x+a)
*a^4*c+2520/b^9*d^8/(b*x+a)*a^3*c^2-2520/b^8*d^7/(b*x+a)*a^2*c^3+1260/b^7*d^6/(b
*x+a)*a*c^4-840/b^10*d^9*ln(b*x+a)*a^3*c+1260/b^9*d^8*ln(b*x+a)*a^2*c^2-840/b^8*
d^7*ln(b*x+a)*a*c^3+5/3/b^10/(b*x+a)^6*a^9*c*d^9-15/2/b^9/(b*x+a)^6*a^8*c^2*d^8+
20/b^8/(b*x+a)^6*a^7*c^3*d^7-35/b^7/(b*x+a)^6*a^6*c^4*d^6+42/b^6/(b*x+a)^6*a^5*c
^5*d^5-35/b^5/(b*x+a)^6*a^4*c^6*d^4+20/b^4/(b*x+a)^6*a^3*c^7*d^3-15/2/b^3/(b*x+a
)^6*a^2*c^8*d^2+5/3/b^2/(b*x+a)^6*a*c^9*d+90/b^10*d^9/(b*x+a)^4*a^7*c-315/b^9*d^
8/(b*x+a)^4*a^6*c^2+630/b^8*d^7/(b*x+a)^4*a^5*c^3-1575/2/b^7*d^6/(b*x+a)^4*a^4*c
^4+630/b^6*d^5/(b*x+a)^4*a^3*c^5-315/b^5*d^4/(b*x+a)^4*a^2*c^6+90/b^4*d^3/(b*x+a
)^4*a*c^7+168/b^5*d^4/(b*x+a)^5*a^3*c^6-72/b^4*d^3/(b*x+a)^5*a^2*c^7+18/b^3*d^2/
(b*x+a)^5*a*c^8-18/b^10*d^9/(b*x+a)^5*a^8*c+72/b^9*d^8/(b*x+a)^5*a^7*c^2-168/b^8
*d^7/(b*x+a)^5*a^6*c^3+252/b^7*d^6/(b*x+a)^5*a^5*c^4-252/b^6*d^5/(b*x+a)^5*a^4*c
^5-35*d^9/b^8*x^2*a*c+280*d^9/b^9*a^2*c*x-315*d^8/b^8*a*c^2*x-1400/b^8*d^7/(b*x+
a)^3*a^4*c^3+1400/b^7*d^6/(b*x+a)^3*a^3*c^4-840/b^6*d^5/(b*x+a)^3*a^2*c^5+280/b^
5*d^4/(b*x+a)^3*a*c^6+630/b^10*d^9/(b*x+a)^2*a^5*c-1575/b^9*d^8/(b*x+a)^2*a^4*c^
2+2100/b^8*d^7/(b*x+a)^2*a^3*c^3-1575/b^7*d^6/(b*x+a)^2*a^2*c^4+630/b^6*d^5/(b*x
+a)^2*a*c^5-280/b^10*d^9/(b*x+a)^3*a^6*c+840/b^9*d^8/(b*x+a)^3*a^5*c^2+1/4*d^10/
b^7*x^4-1/6/b/(b*x+a)^6*c^10

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Maxima [A]  time = 1.48777, size = 1249, normalized size = 4.77 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^10/(b*x + a)^7,x, algorithm="maxima")

[Out]

-1/12*(2*b^10*c^10 + 4*a*b^9*c^9*d + 9*a^2*b^8*c^8*d^2 + 24*a^3*b^7*c^7*d^3 + 84
*a^4*b^6*c^6*d^4 + 504*a^5*b^5*c^5*d^5 - 6174*a^6*b^4*c^4*d^6 + 16056*a^7*b^3*c^
3*d^7 - 18414*a^8*b^2*c^2*d^8 + 10036*a^9*b*c*d^9 - 2131*a^10*d^10 + 3024*(b^10*
c^5*d^5 - 5*a*b^9*c^4*d^6 + 10*a^2*b^8*c^3*d^7 - 10*a^3*b^7*c^2*d^8 + 5*a^4*b^6*
c*d^9 - a^5*b^5*d^10)*x^5 + 1260*(b^10*c^6*d^4 + 6*a*b^9*c^5*d^5 - 45*a^2*b^8*c^
4*d^6 + 100*a^3*b^7*c^3*d^7 - 105*a^4*b^6*c^2*d^8 + 54*a^5*b^5*c*d^9 - 11*a^6*b^
4*d^10)*x^4 + 240*(2*b^10*c^7*d^3 + 7*a*b^9*c^6*d^4 + 42*a^2*b^8*c^5*d^5 - 385*a
^3*b^7*c^4*d^6 + 910*a^4*b^6*c^3*d^7 - 987*a^5*b^5*c^2*d^8 + 518*a^6*b^4*c*d^9 -
107*a^7*b^3*d^10)*x^3 + 45*(3*b^10*c^8*d^2 + 8*a*b^9*c^7*d^3 + 28*a^2*b^8*c^6*d
^4 + 168*a^3*b^7*c^5*d^5 - 1750*a^4*b^6*c^4*d^6 + 4312*a^5*b^5*c^3*d^7 - 4788*a^
6*b^4*c^2*d^8 + 2552*a^7*b^3*c*d^9 - 533*a^8*b^2*d^10)*x^2 + 6*(4*b^10*c^9*d + 9
*a*b^9*c^8*d^2 + 24*a^2*b^8*c^7*d^3 + 84*a^3*b^7*c^6*d^4 + 504*a^4*b^6*c^5*d^5 -
5754*a^5*b^5*c^4*d^6 + 14616*a^6*b^4*c^3*d^7 - 16524*a^7*b^3*c^2*d^8 + 8916*a^8
*b^2*c*d^9 - 1879*a^9*b*d^10)*x)/(b^17*x^6 + 6*a*b^16*x^5 + 15*a^2*b^15*x^4 + 20
*a^3*b^14*x^3 + 15*a^4*b^13*x^2 + 6*a^5*b^12*x + a^6*b^11) + 1/12*(3*b^3*d^10*x^
4 + 4*(10*b^3*c*d^9 - 7*a*b^2*d^10)*x^3 + 6*(45*b^3*c^2*d^8 - 70*a*b^2*c*d^9 + 2
8*a^2*b*d^10)*x^2 + 12*(120*b^3*c^3*d^7 - 315*a*b^2*c^2*d^8 + 280*a^2*b*c*d^9 -
84*a^3*d^10)*x)/b^10 + 210*(b^4*c^4*d^6 - 4*a*b^3*c^3*d^7 + 6*a^2*b^2*c^2*d^8 -
4*a^3*b*c*d^9 + a^4*d^10)*log(b*x + a)/b^11

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Fricas [A]  time = 0.208362, size = 1871, normalized size = 7.14 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^10/(b*x + a)^7,x, algorithm="fricas")

[Out]

1/12*(3*b^10*d^10*x^10 - 2*b^10*c^10 - 4*a*b^9*c^9*d - 9*a^2*b^8*c^8*d^2 - 24*a^
3*b^7*c^7*d^3 - 84*a^4*b^6*c^6*d^4 - 504*a^5*b^5*c^5*d^5 + 6174*a^6*b^4*c^4*d^6
- 16056*a^7*b^3*c^3*d^7 + 18414*a^8*b^2*c^2*d^8 - 10036*a^9*b*c*d^9 + 2131*a^10*
d^10 + 10*(4*b^10*c*d^9 - a*b^9*d^10)*x^9 + 45*(6*b^10*c^2*d^8 - 4*a*b^9*c*d^9 +
a^2*b^8*d^10)*x^8 + 360*(4*b^10*c^3*d^7 - 6*a*b^9*c^2*d^8 + 4*a^2*b^8*c*d^9 - a
^3*b^7*d^10)*x^7 + (8640*a*b^9*c^3*d^7 - 18630*a^2*b^8*c^2*d^8 + 14660*a^3*b^7*c
*d^9 - 4043*a^4*b^6*d^10)*x^6 - 6*(504*b^10*c^5*d^5 - 2520*a*b^9*c^4*d^6 + 1440*
a^2*b^8*c^3*d^7 + 3510*a^3*b^7*c^2*d^8 - 4580*a^4*b^6*c*d^9 + 1523*a^5*b^5*d^10)
*x^5 - 15*(84*b^10*c^6*d^4 + 504*a*b^9*c^5*d^5 - 3780*a^2*b^8*c^4*d^6 + 6480*a^3
*b^7*c^3*d^7 - 4050*a^4*b^6*c^2*d^8 + 460*a^5*b^5*c*d^9 + 263*a^6*b^4*d^10)*x^4
- 20*(24*b^10*c^7*d^3 + 84*a*b^9*c^6*d^4 + 504*a^2*b^8*c^5*d^5 - 4620*a^3*b^7*c^
4*d^6 + 9840*a^4*b^6*c^3*d^7 - 9090*a^5*b^5*c^2*d^8 + 3820*a^6*b^4*c*d^9 - 577*a
^7*b^3*d^10)*x^3 - 15*(9*b^10*c^8*d^2 + 24*a*b^9*c^7*d^3 + 84*a^2*b^8*c^6*d^4 +
504*a^3*b^7*c^5*d^5 - 5250*a^4*b^6*c^4*d^6 + 12360*a^5*b^5*c^3*d^7 - 12870*a^6*b
^4*c^2*d^8 + 6340*a^7*b^3*c*d^9 - 1207*a^8*b^2*d^10)*x^2 - 6*(4*b^10*c^9*d + 9*a
*b^9*c^8*d^2 + 24*a^2*b^8*c^7*d^3 + 84*a^3*b^7*c^6*d^4 + 504*a^4*b^6*c^5*d^5 - 5
754*a^5*b^5*c^4*d^6 + 14376*a^6*b^4*c^3*d^7 - 15894*a^7*b^3*c^2*d^8 + 8356*a^8*b
^2*c*d^9 - 1711*a^9*b*d^10)*x + 2520*(a^6*b^4*c^4*d^6 - 4*a^7*b^3*c^3*d^7 + 6*a^
8*b^2*c^2*d^8 - 4*a^9*b*c*d^9 + a^10*d^10 + (b^10*c^4*d^6 - 4*a*b^9*c^3*d^7 + 6*
a^2*b^8*c^2*d^8 - 4*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 6*(a*b^9*c^4*d^6 - 4*a^2
*b^8*c^3*d^7 + 6*a^3*b^7*c^2*d^8 - 4*a^4*b^6*c*d^9 + a^5*b^5*d^10)*x^5 + 15*(a^2
*b^8*c^4*d^6 - 4*a^3*b^7*c^3*d^7 + 6*a^4*b^6*c^2*d^8 - 4*a^5*b^5*c*d^9 + a^6*b^4
*d^10)*x^4 + 20*(a^3*b^7*c^4*d^6 - 4*a^4*b^6*c^3*d^7 + 6*a^5*b^5*c^2*d^8 - 4*a^6
*b^4*c*d^9 + a^7*b^3*d^10)*x^3 + 15*(a^4*b^6*c^4*d^6 - 4*a^5*b^5*c^3*d^7 + 6*a^6
*b^4*c^2*d^8 - 4*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 6*(a^5*b^5*c^4*d^6 - 4*a^6*
b^4*c^3*d^7 + 6*a^7*b^3*c^2*d^8 - 4*a^8*b^2*c*d^9 + a^9*b*d^10)*x)*log(b*x + a))
/(b^17*x^6 + 6*a*b^16*x^5 + 15*a^2*b^15*x^4 + 20*a^3*b^14*x^3 + 15*a^4*b^13*x^2
+ 6*a^5*b^12*x + a^6*b^11)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)**10/(b*x+a)**7,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.226644, size = 1185, normalized size = 4.52 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^10/(b*x + a)^7,x, algorithm="giac")

[Out]

210*(b^4*c^4*d^6 - 4*a*b^3*c^3*d^7 + 6*a^2*b^2*c^2*d^8 - 4*a^3*b*c*d^9 + a^4*d^1
0)*ln(abs(b*x + a))/b^11 - 1/12*(2*b^10*c^10 + 4*a*b^9*c^9*d + 9*a^2*b^8*c^8*d^2
+ 24*a^3*b^7*c^7*d^3 + 84*a^4*b^6*c^6*d^4 + 504*a^5*b^5*c^5*d^5 - 6174*a^6*b^4*
c^4*d^6 + 16056*a^7*b^3*c^3*d^7 - 18414*a^8*b^2*c^2*d^8 + 10036*a^9*b*c*d^9 - 21
31*a^10*d^10 + 3024*(b^10*c^5*d^5 - 5*a*b^9*c^4*d^6 + 10*a^2*b^8*c^3*d^7 - 10*a^
3*b^7*c^2*d^8 + 5*a^4*b^6*c*d^9 - a^5*b^5*d^10)*x^5 + 1260*(b^10*c^6*d^4 + 6*a*b
^9*c^5*d^5 - 45*a^2*b^8*c^4*d^6 + 100*a^3*b^7*c^3*d^7 - 105*a^4*b^6*c^2*d^8 + 54
*a^5*b^5*c*d^9 - 11*a^6*b^4*d^10)*x^4 + 240*(2*b^10*c^7*d^3 + 7*a*b^9*c^6*d^4 +
42*a^2*b^8*c^5*d^5 - 385*a^3*b^7*c^4*d^6 + 910*a^4*b^6*c^3*d^7 - 987*a^5*b^5*c^2
*d^8 + 518*a^6*b^4*c*d^9 - 107*a^7*b^3*d^10)*x^3 + 45*(3*b^10*c^8*d^2 + 8*a*b^9*
c^7*d^3 + 28*a^2*b^8*c^6*d^4 + 168*a^3*b^7*c^5*d^5 - 1750*a^4*b^6*c^4*d^6 + 4312
*a^5*b^5*c^3*d^7 - 4788*a^6*b^4*c^2*d^8 + 2552*a^7*b^3*c*d^9 - 533*a^8*b^2*d^10)
*x^2 + 6*(4*b^10*c^9*d + 9*a*b^9*c^8*d^2 + 24*a^2*b^8*c^7*d^3 + 84*a^3*b^7*c^6*d
^4 + 504*a^4*b^6*c^5*d^5 - 5754*a^5*b^5*c^4*d^6 + 14616*a^6*b^4*c^3*d^7 - 16524*
a^7*b^3*c^2*d^8 + 8916*a^8*b^2*c*d^9 - 1879*a^9*b*d^10)*x)/((b*x + a)^6*b^11) +
1/12*(3*b^21*d^10*x^4 + 40*b^21*c*d^9*x^3 - 28*a*b^20*d^10*x^3 + 270*b^21*c^2*d^
8*x^2 - 420*a*b^20*c*d^9*x^2 + 168*a^2*b^19*d^10*x^2 + 1440*b^21*c^3*d^7*x - 378
0*a*b^20*c^2*d^8*x + 3360*a^2*b^19*c*d^9*x - 1008*a^3*b^18*d^10*x)/b^28